Three Similar Polygons
The following problem and its generalization have been discussed elsewhere:
Suppose A, B, C are arbitrary points on a straight line and X is a point not on the line. Construct similar and similarly oriented triangles ABX and BCY. If triangle XYZ is similar to triangles ABX and BCY but with a different orientation then Z is always collinear with A, B, and C!
The applet below presents an additional twist. Consider similar polygons: ABX0X1...XN, BCY0Y1...YN, X0Y0Z0...ZN. The problem above implies that if one of the vertices Xi, i > 0, lies on the line X0Y0, then the vertex Zi lies on AB. Perhaps a little more surprising is the fact that if Xi lies on AB then Zi lies on X0Y0, whereas Xi and Yi are always on the same line: either both are on X0Y0 or both are on AB.
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